User:Rjowsey/sandbox

In mathematics, the complexification of a real vector space results in a complex vector space (over the complex number field). To "complexify" a space means that we extend ordinary multiplication by real numbers to include multiplication by complex numbers. In mathematical physics, when we complexify a real coordinate space Rⁿ we create a complex coordinate space Cⁿ, referred to in differential geometry as a "complex manifold".

The Minkowski space of Special and General Relativity (GR) is a "pseudo-euclidean" vector space. The spacetime underlying Einstein's Field Equations, which mathematically describe gravitation, is a 4-dimensional "Reimannian manifold", in which the four dimensions of space and time (x, y, z, ict) are regarded as spatial distances, measured in metres (or light-seconds), and fundamentally interchangeable. Any point in complex spacetime can be represented by a vector in the complex plane; its real position (x) denoted by cos(φ), and its imaginary time (ict) value by sin(φ).

In 1919, Theodor Kaluza posted his 5-dimensional extension of General Relativity to Einstein, who was deeply impressed at the natural way in which the equations of electromagnetism emerged from Kaluza's five-dimensional maths. In 1926, Oskar Klein suggested that Kaluza's extra dimension might be "curled up" into an infinitesimal circle, as if a circular topology is hidden within every point in space. Instead of being another spatial dimension, the extra dimension could be thought of as an angle, which created a hyper-dimension as it spun through 360°.

For several decades after publishing his General Theory of Relativity in 1915, Einstein tried to unify gravitational field theory with Maxwell's electromagnetism, to create a unified field theory which could explain the gravitational and electromagnetic fields in terms of a universal "gravito-electro-magnetic" field. He was attempting to consolidate the fundamental forces and the elementary particles into a single uniform field theory. His last few scientific papers reveal his valiant struggles to unify GR and QM in Kaluza's 5-dimensional spacetime.

"Scientifically, I am still lagging because of the same mathematical difficulties which make it impossible for me to affirm or contradict my more general relativistic field theory [...]. I will not be able to finish it [the work]; It will be forgotten and at a later time arguably mist be re-discovered. It happened this way with so many problems." — Albert Einstein, correspondence with Maurice Solovine, 25 Nov 1948

In 1953, Wolfgang Pauli generalised the Kaluza-Klein theory to a six-dimensional space, and (using dimensional reduction) derived the essentials of an SU(2) gauge theory (applied in QM to the electroweak interaction), as if Klein's "curled up" circle had become the surface of an infinitesimal hyper-sphere.

Although the wavefunctions and particles of quantum mechanics (QM) are thought of as inhabiting the exact-same Minkowski space as described by GR, the (configuration) state space in QM is actually a multi-dimensional, complex (Hilbert) vector space. Imaginary numbers fall from every equation, and behind all the mathematics of QM is Euler's formula, "the most remarkable formula in mathematics" according to Richard Feynman:

$e^{ix}=\cos x+i\sin x$

The exact-same equation defines Kaluza-Klein's extra "curled up" spatial dimension, and it is the basis of the complex numbers, and of the unitary group U(1), also known as the circle group, which is the most fundamental symmetry in the universe.

Deep Dimensional Analysis

History

During the late 1860's, James Clerk Maxwell's ideas about electromagnetism gradually became more mathematically complex. His spacetime geometry contained two imaginary dimensions to accommodate the electric potential E and the magnetic field H, plus another imaginary dimension for the gravitational potential V, all three being mathematically orthogonal to real Euclidean space. In his discussion of the findings of the great electromagnetic experimentalist Michael Faraday, these comprised six spatial dimensions (three real and three imaginary).

"I am getting converted to Quaternions, and have put some in my book, in a heretical form..." — James Clerk Maxwell, correspondence with Prof. Lewis Campbell, 19 Oct 1872

In his scientific description of electromagnetism, Maxwell used what he called a "heretical form" of quaternion algebra, which explicitly separated the three imaginary dimensions from the real part. He stated emphatically that tensors and vectors were inadequate mathematical tools to correctly encapsulate the electromagnetic fields and forces. He also quietly discussed with colleagues how one might detect and measure "non-observable" or "hidden" spatial dimensions, which he conceived of as "storing energy", both kinetic and potential, in the elastic fabric of space itself.

"The peculiarity of our space is that of its three dimensions, none is before or after another. As is x, so is y, and so is z. If you have 4 dimensions, this becomes a puzzle. For first, if three of them are in our space, then which three? Also, if we lived in space of m dimensions, but were only capable of thinking n of them, then first, which n? Second, if so, things would happen requiring the rest to explain them, and so we should either be stultified or made wiser. I am quite sure that the kind of continuity which has four dimensions all co-equal, is not to be discovered by merely generalising Cartesian space equations." — James Clerk Maxwell, in correspondence with C.J. Monro, Esq., 15 Mar 1871

James Clerk Maxwell (1831–1879)

His preferred quaternion notation was eliminated from A Treatise on Electricity and Magnetism at the insistence of his publisher (over his strenuous objections), because very few scientists at the time could understood the maths. Maxwell regrettably passed away in 1879 at age 48, when he was only partway through his revision for the second edition.

Maxwell played a major role in establishing the modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived. Although he defined length, time and mass to be "the three fundamental units", he also noted that gravitational mass can be derived from length and time by assuming a form of Newton's law of universal gravitation in which the gravitational constant G is taken as unity, giving M = L³T⁻².

By assuming a form of Coulomb's law in which Coulomb's constant k_e is taken as unity, Maxwell then determined that the dimensions of an electrostatic unit of charge were Q = L^3/2M^1/2T⁻¹, which, after substituting his M = L³T⁻² equation for mass, results in charge having the same fundamental dimensions as mass, viz. Q = L³T⁻². This peculiarity must have intrigued and puzzled him, but he apparently never discussed it in his lectures, correspondence or scientific writings.

The fundamental physical constants

The primary physical constant is the speed of light in vacuum (c₀), which has unitary space-time dimensions of L/T, i.e. metres per second.

Another fundamental is Newton's gravitational constant (G), with SI dimensions of L³/T²M. When Maxwell's dimensions for mass (M = L³/T²) are substituted into the SI dimensions, the gravitational constant is shown to be dimensionless in elemental space-time, viz. L⁰/T⁰.

Planck's constant (h), the fundamental ratio of a quantum of energy to its wavefunction's frequency (T⁻¹), has SI dimensions of L²M/T. Substituting Maxwell's dimensions for mass (M=L³/T²) shows that the Planck quantum of Action has fundamental dimensions of L⁵/T³.

Maxwell determined that the unit of elementary charge (e) has dimensions of (L³M/T²)^½. Substituting his mass dimensions (M = L³/T²) reveals that charge has the fundamental dimensions of (L⁶/T⁴)^1/2, i.e. Q = L³/T².

The Boltzmann constant (k_B) is defined as the energy in Joules per degree of temperature (Θ), having SI dimensions of L²M/T²Θ. Substituting M = L³/T² reveals the Boltzmann constant to have space-time dimensions of L⁵/T⁴ (energy) per degree K.

Universal physical constants normalized in 2 dimensions (L,T)

Constant	Symbol	SI Dimensions	L/T Dimensions
Speed of light in vacuum	c0	LT−1	LT−1
Gravitational constant	G	L3M−1 T−2	L0T0
Planck constant	h	L2MT−1	L5T−3
Coulomb constant	ke	L3MT−2Q−2	L0T0
Boltzmann constant	kB	L2MT−2Θ−1	L5T−4Θ−1

The Planck units

The Planck Units are "natural units" of measurement defined exclusively in terms of five universal physical constants, viz. c, G, ħ, k_e and k_B, such that these constants have the numerical value of 1 when expressed in terms of the Planck units.

The base spatial unit is the Planck length (ℓ_P), defined as the distance traveled by light in vacuum during one Planck time (t_P). The numerical value of ℓ_P is calculated from (ħGc⁻³)^1/2, the fundamental space-time dimensions of which resolve as (L⁵T⁻³⋅L⁻³T³)^1/2 = L.

The five base Planck units, viz. length, time, mass, charge and temperature, have traditionally been dimensioned in terms of the base SI units L, T, M, Q and Θ. However, Maxwell's factoring of mass and charge into the more fundamental space-time dimensions of L³T⁻² permits a deep two-dimensional analysis of the base and derived Planck units.

Since the gravitational constant G and the vacuum permittivity (electric constant) ε₀ are dimensionless in L/T space-time basis units, they can be factored out of the Planck units, thereby simplifying the dimensional analysis. For example, Planck area is defined as ħG/c³, which simplifies to L⁵T⁻³ ⋅ L⁻³T³ = L². Similarly, Planck current is defined as (4πε₀c⁶/G)^1/2, which resolves to (L⁶T⁻⁶)^1/2 = L³T⁻³.

Thus, the fundamental space-time dimensions for each of the Planck units can be derived from their defining expressions. However, it is considerably easier to simply substitute L³T⁻² for M and Q in the conventional SI dimensions of the Planck quantities, as follows:

The Planck units normalized in 2 dimensions (L,T)

Name	SI Dimensions	L/T Dimensions
Planck length	L	L
Planck time	T	T
Planck area	L2	L2
Planck volume	L3	L3
Planck mass	M	L3T−2
Planck charge	Q	L3T−2
Planck momentum	LMT−1	L4T−3
Planck force	LMT−2	L4T−4
Planck energy	L2MT−2	L5T−4
Planck power	L2MT−3	L5T−5
Planck density	ML−3	T−2
Planck intensity	MT−3	L3T−5
Planck frequency	T−1	T−1
Planck pressure	L−1MT−2	L2T−4
Planck current	QT−1	L3T−3
Planck voltage	L2MT−2Q−1	L2T−2
Planck resistance	L2MT−1Q−2	L−1T

To facilitate further analysis, these quantities can be arranged into a log-log space/time matrix, whose columns represent incrementing powers of Planck length (Lⁿ) and whose rows represent increasing powers of inverse-time (T^−m):

The Planck units arranged in a 6D space/time matrix

L/T	L0	L1	L2	L3	L4	L5
T0	$G{,}$ $\epsilon$0 Constants	${\sqrt {\frac {\hbar }{c^{3}}}}$ Length	${\frac {\hbar }{c^{3}}}$ Surface Area	${\sqrt {\frac {\hbar ^{3}}{c^{9}}}}$ Volume	${\frac {\hbar ^{2}}{c^{6}}}$ 4D-Volume	${\sqrt {\frac {\hbar ^{5}}{c^{15}}}}$ 5D-Volume
T−1	${\sqrt {\frac {c^{5}}{\hbar }}}$ Frequency	$c$ Velocity	${\sqrt {\frac {\hbar }{c}}}$ Magnetic flux	${\frac {\hbar }{c^{2}}}$ Flow rate	${\sqrt {\frac {\hbar ^{3}}{c^{7}}}}$	${\frac {\hbar ^{2}}{c^{5}}}$
T−2	${\frac {c^{5}}{\hbar }}$ Mass density	${\sqrt {\frac {c^{7}}{\hbar }}}$ Acceleration	$c^{2}$ Voltage	${\sqrt {\hbar c}}$ Mass, Charge	${\frac {\hbar }{c}}$	${\sqrt {\frac {\hbar ^{3}}{c^{5}}}}$
T−3	${\sqrt {\frac {c^{15}}{\hbar ^{3}}}}$	${\frac {c^{6}}{\hbar }}$ Current flux	${\sqrt {\frac {c^{9}}{\hbar }}}$	$c^{3}$ Current	${\sqrt {\hbar c^{3}}}$ Momentum	$\hbar$ Planck Action
T−4	${\frac {c^{10}}{\hbar ^{2}}}$	${\sqrt {\frac {c^{17}}{\hbar ^{3}}}}$	${\frac {c^{7}}{\hbar }}$ Pressure	${\sqrt {\frac {c^{11}}{\hbar }}}$	$c^{4}$ Force	${\sqrt {\hbar c^{5}}}$ Energy
T−5	${\sqrt {\frac {c^{25}}{\hbar ^{5}}}}$	${\frac {c^{11}}{\hbar ^{2}}}$	${\sqrt {\frac {c^{19}}{\hbar ^{3}}}}$	${\frac {c^{8}}{\hbar }}$ Intensity	${\sqrt {\frac {c^{13}}{\hbar }}}$	$c^{5}$ Power

Five mutually-orthogonal spatial dimensions are required to accommodate all the Planck units, notably the "higher dimensional" (L⁴, L⁵) quantities of momentum, force, action, energy and power. Three of the spatial dimensions are the real linear dimensions of Euclidean x,y,z space, viz. length, breadth and height. Like the "time dimension" of special relativity, defined by Einstein (1905) as √-1∙c∙t, the two extra spatial dimensions are mathematically imaginary by virtue of their orthogonality, i.e. being Wick-rotated relative to all the other dimensions. The space/time matrix has internal symmetries corresponding to the SU(3) × SU(2) × U(1) unitary group, consistent with the Standard Model.

In 2006, Paul Wesson determined that an extra spatial coordinate x⁴ could be identified as ℓ = Gm/c², which he termed the "Einstein gauge". Formulated in terms of momentum, i.e. ℓ = Gp/c³, this gauge corresponds to the L⁴ spatial dimension of the space/time matrix, from which emerges momentum, force, and pressure. Wesson also identified another spatial coordinate as ℓ = ħ/mc (dimensionally identical to ℓ = ħ/qc), which he termed the "Planck gauge". This ħ/qc gauge corresponds to the L⁵ spatial coordinate in the space/time matrix. The physical quantities of action, energy, power and intensity emerge from this imaginary dimension.

Electromagnetism

Substitution of Maxwell's L³T⁻² for mass (M) and charge (Q) in the SI dimensions of the electromagnetic (EM) quantities effects a "flattening" of their dimensionality to just spatial length and time, as follows:

The electromagnetic units in 2 space/time dimensions (L, T)

Quantity	Symbol	Unit	SI Dimensions	L/T Dimensions
Electric Charge	Q	coulomb	Q	L3T−2
Electric Current	I	ampere	QT−1	L3T−3
Electric Potential	Δ$\varphi$	volt	L2MT−2Q−1	L2T−2
Magnetic Field Strength	H	ampere per metre	QL−1T−1	L2T−3
Electric Displacement	D	coulomb per metre2	QL−2	LT−2
Electric Field Strength	E	volt per metre	LMT−2Q−1	LT−2
Current Density	J	ampere per metre2	QT−1L−2	LT−3
Charge Density	ρq	coulomb per metre3	QL−3	T−2
Electric Resistance	R	ohm	L2MT−1Q−2	L−1T
Capacitance	C	farad	T2Q2L−2M−1	L
Permittivity	ε	farad per metre	T2Q2L−3M−1	1
Magnetic Flux Density	B	tesla	MT−1Q−1	T−1
Magnetic Potential	A	weber per metre	LMT−1Q−1	LT−1
Magnetic Flux	Φ	weber	L2MT−1Q−1	L2T−1
Energy Density	ρe	joule per metre3	ML−1T−2	L2T−4
Inductance	L	henry	ML2Q−2	L−1T2
Permeability	μ	henry per metre	MLQ−2	L−2T2
Magnetic Momentum	p	newton-second	LMT−1	L4T−3
Electromagnetic Force	F	newton	LMT−2	L4T−4
Electromagnetic Action	S	joule-second	L2MT−1	L5T−3
Electric Potential Energy	U	joule	L2MT−2	L5T−4
Radiant EM Energy	E	joule	L2MT−2	L5T−4
Poynting Vector	S	watt per metre2	MT−3	L3T−5
Power	P	watt	L2MT−3	L5T−5

As with the Planck Units, these quantities can be arranged in a 6-dimensional space/time matrix, with columns representing powers of Planck length (Lⁿ), and rows which represent powers of inverse-time (T^−m). Noting that the differential (or gradient) of any quantity with respect to time (d/dt) is the unit immediately below it, and that the differential of a quantity with respect to space (d/ds or ∇) is the unit to its left, Maxwell's equations can be discerned in the relationships between these electromagnetic quantities. Within the log-log matrix, multiplication is effected by adding the two units' dimensional indices, e.g. L^aT^b × L^cT^d = L^(a+c)T^(b+d), and division is performed by subtracting the denominator's space/time indices from the numerator's.

The electromagnetic quantities arranged in a 6D space/time matrix

L/T	L0	L1	L2	L3	L4	L5
T0	Electric constant $\epsilon$0 = 1	Wavelength λ Capacitance C	Surface area	Volume	4D-volume	5D-volume
T−1	Magnetic flux density B	Magnetic vector potential A	Magnetic flux Φ	∫ Φ⋅ds ∫ Q dt
T−2	Charge density ρq = ∇ ⋅ E	Electric field strength E (D)	Electrical potential Δ$\varphi$	Charge Q	∫ Q⋅ds ∫ p dt	∫ S⋅dt
T−3		Current density J = ∇ × H	Magnetic field strength H	Current I	Momentum p = Q⋅A	Action S
T−4			Energy density ρe = B ⋅ H	∇F = A × H	Force F = Q (E + v x B)	Energy Uε = Q⋅Δ$\varphi$
T−5		∇2S	∇ ⋅ S	Poynting vector S = E × H	dF/dt = ∇P	Power P = Δ$\varphi$ ⋅ I

Imaginary Dimensions

Imaginary spatial dimensions in complex 6D space/time may be formulated using various equivalent expressions, all of which resolve to the dimension of spatial distance (L).

The "Einstein gauge" is canonically formulated as Gm/c² (half the Schwarzschild radius), but is more usefully expressed in terms of momentum, viz. ℓ = p/c³, particularly in the context of 6-dimensional Special Relativity. It can also be expressed in terms of kinetic energy as ℓ= E_k/c⁴. In the context of electromagnetism, this metric is best formulated in terms of magnetic momentum divided by current (flow of charge): ℓ = p/I. In quantum mechanics, this imaginary dimension can be formulated in terms of a wavefunction's frequency of oscillation: ℓ = iħƒ/c⁴ (where frequency ƒ = c/λ and energy E = ħƒ).

The "Planck gauge" was originally formulated as ħ/mc in the context of higher-dimensional (Kaluza–Klein) gravitation. In electromagnetism, this imaginary dimension may be equivalently expressed as ℓ = ħ/qc = q/c². In regards to electric potential (voltage), it is formulated by ℓ = q/V = ħc/qV, and in quantum mechanics it is most usefully expressed in terms of potential energy, viz. ℓ = iħc/E₀ = iq²/E₀ (where E₀ = mc² = qV).

The 5th spatial dimension is associated with potential energy and inverse-time, analogous to the 4th imaginary dimension's association with spatial position and momentum (or kinetic energy). The complex dimensions associated with potential and kinetic energy are clearly orthogonal, since total energy squared is given by E² = E₀² + E_k². Heisenburg's Uncertainty Principle can also be expressed in terms of position and momentum, as σ_xσ_p ≥ ħ/2, or in terms of potential energy and time, viz. σ_uσ_t ≥ ħ/2.

Special Relativity

The animated graphic below simulates a test particle with Planck mass being accelerated to the speed of light, where it has Planck momentum and Planck kinetic energy. The simplified Minkowski diagram at the top-left shows the "4D Lorentz rotation" of the moving frame of reference inhabited by the particle. The bottom-left L⁴ projection shows length contraction in the moving frame of reference, with the real spatial dimension (ʀ) plotted against the "Einstein gauge" representing imaginary momentum, having the canonical unit of Gp/c³ in hyperspatial Special Relativity. The top-right L⁵ projection shows time dilation approaching infinity at light speed; imaginary time ict is plotted against the "Planck gauge", the imaginary x-axis unit canonically defined by λᵩ = ħ/mc, the Compton wavelength.

<\p>

The Lorentz factor is the inverse-cosine of the phase angle (0 < φ < ^π/₂), i.e. γ = 1/cos(φ), and the ratio of the particle's velocity to light speed is β = ^v/_c = sin(φ). Thus, time dilation and length contraction simplify to τᵩ = t∙cos(φ) and ʀᵩ = r∙cos(φ). The y-axis in the bottom-right projection represents the imaginary component of the particle's kinetic energy, while the x-axis represents the imaginary component of its potential energy (mc²) in units of Planck energy (E_P). The particle's total energy Eᵩ is a function of √((t∙sin(φ))² + (r∙sin(φ))²). At light speed, the particle's rest-mass energy and its momentum can be seen as inhabiting two imaginary spatial dimensions. A Planck mass moving at the velocity of light has total energy of √2∙E_P (assuming there is no energy loss due to gravitational radiation). The particle's matter-wave has a de Broglie wavelength λᵩ of one Planck length, and its wavefunction is oscillating at Planck frequency.

See also

• Spacetime
• Quantum gravity
• Quantum electrodynamics
• AdS/CFT correspondence
• Holographic principle
• Bargmann–Wigner equations
• Biquaternions
• Complex bivectors
• Poincaré group

References

Selected papers

• "On the Electrodynamics of Moving Bodies". Translation by George Barker Jeffery and Wilfrid Perrett in The Principle of Relativity, London: Methuen and Company, Ltd. (1923)
• J. B. Hartle, S. W. Hawking (1983). "Wave function of the Universe". Physical Review D. 28 (12): 2960. Bibcode:1983PhRvD..28.2960H. doi:10.1103/PhysRevD.28.2960.
• N. Straumann (2000). "On Pauli's invention of non-abelian Kaluza-Klein Theory in 1953". arXiv:gr-qc/0012054v1. {{cite journal}}: Cite journal requires |journal= (help)
• L. Susskind; E. Witten (1998). "The Holographic Bound in Anti-de Sitter Space". arXiv:hep-th/9805114v1. {{cite journal}}: Cite journal requires |journal= (help)
• Juan M. Maldacena (1997). "The Large N Limit of Superconformal Field Theories and Supergravity". arXiv:hep-th/9711200v3. {{cite journal}}: Cite journal requires |journal= (help)
• Jens L. Petersen (1999). "Introduction to the Maldacena Conjecture on AdS/CFT". arXiv:hep-th/9902131v2. {{cite journal}}: Cite journal requires |journal= (help)
• S. W. Hawking; Don N. Page (1982). "Thermodynamics of black holes in anti-de Sitter space". Comm. Math. Phys. 87 (4): 577–588.
• David Hestenes (1990). "The Zitterbewegung Interpretation of Quantum Mechanics" (PDF). Found. Physics. 20 (10): 1213–-1232.
• A. Garrett Lisi (2007). "An Exceptionally Simple Theory of Everything". {{cite journal}}: Cite journal requires |journal= (help)
• J. M. Overduin; P. S. Wesson (2008). "Kaluza-Klein Gravity" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
• M. J. Duff (1994). "Kaluza-Klein Theory in Perspective" (PDF). {{cite journal}}: Cite journal requires |journal= (help)
• Newman, E.T (1973). "Maxwell's equations and complex Minkowski space". J. Math. Phys. 14 (1): 102.
• Albert Einstein (2014). The Meaning of Relativity. Princeton University Press. ISBN 9780691164083.

Further reading

• Richard P. Feynman (1970). The Feynman Lectures on Physics Vol II. Addison Wesley Longman. ISBN 978-0-201-02115-8.
• Landau, L.D.; Lifshitz, E.M. (1951). The classical theory of fields. Addison Wesley Longman.
• Stephen Hawking (1988). A Brief History of Time (PDF). Bantam Dell Publishing Group. ISBN 978-0-553-10953-5. OCLC 39256652.

External links

• Kaluza-Klein Theory
• Anti-de Sitter Space
• The Beginning of Time
• A Geometric Theory of Everything
• A General Systems Approach to a Theory of Everything

Category:Physics

Category:Spacetime

Category:Dimensional analysis